Let's break this problem into parts, starting with finding the intercepts and asymptotes of the rational function:

$r(x) = \frac{2x^2 + 16x - 18}{x^2 + x - 6}$

1. Intercepts

a) x-intercepts

The x-intercepts occur when the numerator of the function is equal to 0. Set the numerator equal to zero and solve for $x$:

$2x^2 + 16x - 18 = 0$

We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula.

Let's start by factoring the equation. First, factor out a 2:

$2(x^2 + 8x - 9) = 0$

Now, factor the quadratic inside the parentheses:

$x^2 + 8x - 9 = (x + 9)(x - 1)$

Thus, the x-intercepts are at $x = -9$ and $x = 1$.

b) y-intercept

The y-intercept occurs when $x = 0$. Substitute $x = 0$ into the function:

$r(0) = \frac{2(0)^2 + 16(0) - 18}{(0)^2 + (0) - 6} = \frac{-18}{-6} = 3$

So, the y-intercept is $(0, 3)$.

2. Asymptotes

a) Vertical Asymptotes

Vertical asymptotes occur where the denominator is equal to zero, provided the numerator is non-zero at those points. Set the denominator equal to zero and solve for $x$:

$x^2 + x - 6 = 0$

Factor the quadratic equation:

$(x - 2)(x + 3) = 0$

So, the vertical asymptotes are at $x = 2$ and $x = -3$.

b) Horizontal Asymptote

For horizontal asymptotes, we consider the degrees of the numerator and denominator. In this case, both the numerator and denominator have degree 2 (since both are quadratic functions). The horizontal asymptote is determined by the ratio of the leading coefficients.

The leading coefficient of the numerator is 2, and the leading coefficient of the denominator is 1. Therefore, the horizontal asymptote is:

$y = \frac{2}{1} = 2$

Summary of Results:

• x-intercepts: $x = -9$ and $x = 1$
• y-intercept: $(0, 3)$
• Vertical asymptotes: $x = 2$ and $x = -3$
• Horizontal asymptote: $y = 2$

Let me know if you'd like a more detailed explanation of any part or have further questions!

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Here are some related questions to deepen your understanding:

1. How do you apply the quadratic formula to solve the equation $2x^2 + 16x - 18 = 0$?
2. What happens if the degree of the numerator and denominator are not equal? How would this affect the horizontal asymptote?
3. How can you determine if a point of discontinuity is a hole or an asymptote in rational functions?
4. What is the process for finding vertical asymptotes in rational functions with more complex denominators?
5. How do you graph rational functions and accurately represent intercepts and asymptotes?

Tip: When factoring quadratics, always check for a common factor first (like we did with the 2 in the numerator). This can simplify the process.